Solving trigonometric equations on stated intervals | Year 11

01 / LEARN

The idea, step by step

Use the full interval

Inverse sine or cosine gives one principal answer. Use unit-circle symmetry and the period to find all candidates, then apply the requested endpoints. For an argument such as 2 x, transform the interval to an interval for 2 x before solving. Divide every resulting angle by 2 at the end.

Do algebra first

If the equation is quadratic in sine or cosine, let u represent the ratio and solve the algebraic equation. Discard any value outside[−1,1]. Avoid dividing by sinx or cosx without checking their zeros, because division could lose roots. Substitute final solutions into the original rule.

A common trap

Solving the transformed argument over the original interval.

Instead: Transform the interval along with the variable; otherwise some cycles may be lost.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Solve $sin x = \sqrt{3} / 2$ on $0 \leq x \leq 2 π$ .

The reference angle isπ/3.
Sine is positive in quadrantsI andII.
Both values lie in the interval.

x = π / 3, 2 π / 3

Example 2

Worked example 2

Solve $cos (2 x) = 1 / 2$ on $0 \leq x \leq 2 π$ .

Let u=2 x, so $0 \leq u \leq 4 π$ .
$u = π / 3, 5 π / 3, 7 π / 3, 11 π / 3$ .
Divide all four values by 2.

x = π / 6, 5 π / 6, 7 π / 6, 11 π / 6

Check your understanding

Solve cosx=−1 on $0 \leq x \leq 2 π$ .

Show answer

x=π.

The cosine minimum occurs once in this interval.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Solve sinx=1/2 on $0 \leq x \leq 2 π$ .
3 marks
Solve $cos x = - \sqrt{2} / 2$ on $0 \leq x \leq 2 π$ .
3 marks
Solve tanx=1 on $- π \leq x \leq π$ .
3 marks
Solve sin(2 x)=0 on $0 \leq x \leq 2 π$ .
3 marks
Solve $2 {sin}^{2} x - sin x = 0$ on $0 \leq x < 2 π$ .
3 marks
Solve $cos (x - π / 4) = 0$ on $0 \leq x \leq 2 π$ .
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Solve sinx=1/2 on $0 \leq x \leq 2 π$ .
3 marks
Use positive sine in quadrantsI andII.
$π / 6, 5 π / 6$
Solve $cos x = - \sqrt{2} / 2$ on $0 \leq x \leq 2 π$ .
3 marks
Use referenceπ/4 in quadrantsII andIII.
$3 π / 4, 5 π / 4$
Solve tanx=1 on $- π \leq x \leq π$ .
3 marks
Tangent repeats afterπ.
$- 3 π / 4, π / 4$
Solve sin(2 x)=0 on $0 \leq x \leq 2 π$ .
3 marks
2 x runs from 0 to 4π; its zeros are multiples ofπ.
$0, π / 2, π, 3 π / 2, 2 π$
Solve $2 {sin}^{2} x - sin x = 0$ on $0 \leq x < 2 π$ .
3 marks
$sin x (2 sin x - 1) = 0$ .
Combine the sinx=0 and sinx=1/2 roots; exclude 2π.
$0, π / 6, 5 π / 6, π$
Solve $cos (x - π / 4) = 0$ on $0 \leq x \leq 2 π$ .
3 marks
$x - π / 4 = π / 2 + k π$ ; retain the two allowed values.
$3 π / 4, 7 π / 4$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

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